1

The question is to test whether $$\int^\infty_1\frac{x^{2012}-20x^7-14}{x^{2014}-30x^{11}+13}dx$$convergent or not. I have found that this rational function which is integrated from 2 to infinity is convergent, and I also founded that there is only one root $\alpha$ between 1 and 2 such that $\alpha^{2014}-30\alpha^{11}+13=0$.

After this, I use Wolfram Alpha to see this graph, and I was surprising that this improper integral is convergent.But I still have no idea how to test.

Now my question is I can not decide whether $$\int^\alpha_1\frac{x^{2012}-20x^7-14}{x^{2014}-30x^{11}+13}dx$$is convergent or not,

Thanks in advance for any help.

This is the situation of the integral near the root of $x^{2014}-30x^{11}+13$, Thanks to Cesareo. The integral from 1.0012 to 1.0016.

Community
  • 1
spencer Dai
  • 11
  • The integral you chose to test it against is divergent, so it does make sense that convergent/divergent $= 0$, but this is not a sufficient argument. You'll have to find an integral that does converge to compare it to. – Ninad Munshi Dec 31 '19 at 09:14
  • Try in Wolfram Alpha int((x^2012 - 20 x^7 - 14)/(x^2014 - 30 x^11 + 13)) from 1.0012 to 1.0016 – Cesareo Dec 31 '19 at 09:44
  • Thanks for NinadMunshi and Cesareo's help, and I rewrite my question according to your comments. Because this integral is convergent, I cannot use a divergent improper integral to compare with this integral. I will try more functions to compare with this question. – spencer Dai Jan 01 '20 at 03:06

0 Answers0